Library iris.program_logic.lifting

The "lifting lemmas" in this file serve to lift the rules of the operational semantics to the program logic.

From iris.proofmode Require Import tactics.
From iris.program_logic Require Export weakestpre.
Set Default Proof Using "Type".

Section lifting.
Context `{!irisG Λ Σ}.
Implicit Types s : stuckness.
Implicit Types v : val Λ.
Implicit Types e : expr Λ.
Implicit Types σ : state Λ.
Implicit Types P Q : iProp Σ.
Implicit Types Φ : val Λ iProp Σ.

Hint Resolve reducible_no_obs_reducible : core.

Lemma wp_lift_step_fupd s E Φ e1 :
  to_val e1 = None
  ( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E,}=∗
    if s is NotStuck then reducible e1 σ1 else True
     e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs ={}=∗ |={,E}=>
      state_interp σ2 κs (length efs + n)
      WP e2 @ s; E {{ Φ }}
      [∗ list] ef efs, WP ef @ s; {{ fork_post }})
   WP e1 @ s; E {{ Φ }}.
Proof. by rewrite wp_unfold /wp_pre=>->. Qed.

Lemma wp_lift_stuck E Φ e :
  to_val e = None
  ( σ κs n, state_interp σ κs n ={E,}=∗ stuck e σ)
   WP e @ E ?{{ Φ }}.
Proof.
  rewrite wp_unfold /wp_pre=>->. iIntros "H" (σ1 κ κs n) "Hσ".
  iMod ("H" with "Hσ") as %[? Hirr]. iModIntro. iSplit; first done.
  iIntros (e2 σ2 efs ?). by case: (Hirr κ e2 σ2 efs).
Qed.

Derived lifting lemmas.
Lemma wp_lift_step s E Φ e1 :
  to_val e1 = None
  ( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E,}=∗
    if s is NotStuck then reducible e1 σ1 else True
     e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs ={,E}=∗
      state_interp σ2 κs (length efs + n)
      WP e2 @ s; E {{ Φ }}
      [∗ list] ef efs, WP ef @ s; {{ fork_post }})
   WP e1 @ s; E {{ Φ }}.
Proof.
  iIntros (?) "H". iApply wp_lift_step_fupd; [done|]. iIntros (????) "Hσ".
  iMod ("H" with "Hσ") as "[$ H]". iIntros "!> * % !> !>". by iApply "H".
Qed.

Lemma wp_lift_pure_step_no_fork `{!Inhabited (state Λ)} s E E' Φ e1 :
  ( σ1, if s is NotStuck then reducible e1 σ1 else to_val e1 = None)
  ( κ σ1 e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs κ = [] σ2 = σ1 efs = [])
  (|={E}[E']▷=> κ e2 efs σ, prim_step e1 σ κ e2 σ efs WP e2 @ s; E {{ Φ }})
   WP e1 @ s; E {{ Φ }}.
Proof.
  iIntros (Hsafe Hstep) "H". iApply wp_lift_step.
  { specialize (Hsafe inhabitant). destruct s; eauto using reducible_not_val. }
  iIntros (σ1 κ κs n) "Hσ". iMod "H".
  iMod fupd_intro_mask' as "Hclose"; last iModIntro; first by set_solver. iSplit.
  { iPureIntro. destruct s; done. }
  iNext. iIntros (e2 σ2 efs ?).
  destruct (Hstep κ σ1 e2 σ2 efs) as (-> & <- & ->); auto.
  iMod "Hclose" as "_". iMod "H". iModIntro.
  iDestruct ("H" with "[//]") as "H". simpl. iFrame.
Qed.

Lemma wp_lift_pure_stuck `{!Inhabited (state Λ)} E Φ e :
  ( σ, stuck e σ)
  True WP e @ E ?{{ Φ }}.
Proof.
  iIntros (Hstuck) "_". iApply wp_lift_stuck.
  - destruct(to_val e) as [v|] eqn:He; last done.
    rewrite -He. by case: (Hstuck inhabitant).
  - iIntrosκs n) "_". by iMod (fupd_intro_mask' E ) as "_"; first set_solver.
Qed.

Lemma wp_lift_atomic_step_fupd {s E1 E2 Φ} e1 :
  to_val e1 = None
  ( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E1}=∗
    if s is NotStuck then reducible e1 σ1 else True
     e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs ={E1}[E2]▷=∗
      state_interp σ2 κs (length efs + n)
      from_option Φ False (to_val e2)
      [∗ list] ef efs, WP ef @ s; {{ fork_post }})
   WP e1 @ s; E1 {{ Φ }}.
Proof.
  iIntros (?) "H".
  iApply (wp_lift_step_fupd s E1 _ e1)=>//; iIntros (σ1 κ κs n) "Hσ1".
  iMod ("H" $! σ1 with "Hσ1") as "[$ H]".
  iMod (fupd_intro_mask' E1 ) as "Hclose"; first set_solver.
  iIntros "!>" (e2 σ2 efs ?). iMod "Hclose" as "_".
  iMod ("H" $! e2 σ2 efs with "[#]") as "H"; [done|].
  iMod (fupd_intro_mask' E2 ) as "Hclose"; [set_solver|]. iIntros "!> !>".
  iMod "Hclose" as "_". iMod "H" as "($ & HQ & $)".
  destruct (to_val e2) eqn:?; last by iExFalso.
  iApply wp_value; last done. by apply of_to_val.
Qed.

Lemma wp_lift_atomic_step {s E Φ} e1 :
  to_val e1 = None
  ( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E}=∗
    if s is NotStuck then reducible e1 σ1 else True
     e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs ={E}=∗
      state_interp σ2 κs (length efs + n)
      from_option Φ False (to_val e2)
      [∗ list] ef efs, WP ef @ s; {{ fork_post }})
   WP e1 @ s; E {{ Φ }}.
Proof.
  iIntros (?) "H". iApply wp_lift_atomic_step_fupd; [done|].
  iIntros (????) "?". iMod ("H" with "[$]") as "[$ H]".
  iIntros "!> *". iIntros (Hstep) "!> !>".
  by iApply "H".
Qed.

Lemma wp_lift_pure_det_step_no_fork `{!Inhabited (state Λ)} {s E E' Φ} e1 e2 :
  ( σ1, if s is NotStuck then reducible e1 σ1 else to_val e1 = None)
  ( σ1 κ e2' σ2 efs', prim_step e1 σ1 κ e2' σ2 efs'
    κ = [] σ2 = σ1 e2' = e2 efs' = [])
  (|={E}[E']▷=> WP e2 @ s; E {{ Φ }}) WP e1 @ s; E {{ Φ }}.
Proof.
  iIntros (? Hpuredet) "H". iApply (wp_lift_pure_step_no_fork s E E'); try done.
  { naive_solver. }
  iApply (step_fupd_wand with "H"); iIntros "H".
  iIntros (κ e' efs' σ (_&?&->&?)%Hpuredet); auto.
Qed.

Lemma wp_pure_step_fupd `{!Inhabited (state Λ)} s E E' e1 e2 φ n Φ :
  PureExec φ n e1 e2
  φ
  (|={E}[E']▷=>^n WP e2 @ s; E {{ Φ }}) WP e1 @ s; E {{ Φ }}.
Proof.
  iIntros (Hexec Hφ) "Hwp". specialize (Hexec Hφ).
  iInduction Hexec as [e|n e1 e2 e3 [Hsafe ?]] "IH"; simpl; first done.
  iApply wp_lift_pure_det_step_no_fork.
  - intros σ. specialize (Hsafe σ). destruct s; eauto using reducible_not_val.
  - done.
  - by iApply (step_fupd_wand with "Hwp").
Qed.

Lemma wp_pure_step_later `{!Inhabited (state Λ)} s E e1 e2 φ n Φ :
  PureExec φ n e1 e2
  φ
  ▷^n WP e2 @ s; E {{ Φ }} WP e1 @ s; E {{ Φ }}.
Proof.
  intros Hexec ?. rewrite -wp_pure_step_fupd //. clear Hexec.
  induction n as [|n IH]; by rewrite //= -step_fupd_intro // IH.
Qed.
End lifting.